Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms AB & BC
L'Hospital's Rule replaces $\lim \frac{f}{g}$ with $\lim \frac{f'}{g'}$, but only after substitution has confirmed the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It differentiates the numerator and denominator separately, which is not the quotient rule, and it may be applied again as long as the form still qualifies, so $\frac{1-\cos x}{x^2}$ takes two rounds to reach $\frac{1}{2}$. Products, differences, and exponential forms have to be rewritten as quotients first.
Two failures dominate. The rule is used without checking, so a determinate form such as $\frac{0}{1}$ gets differentiated and returns a confident wrong number, or the quotient rule is applied by mistake, or one application is treated as enough while the form is still indeterminate. And the result is misread: the number the rule produces is the limit of the original quotient, not a value of the function at the point and not the derivative of the quotient.
The work
3 ways in · any order
Lesson
Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
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Treats the indeterminate form as a precondition to be verified rather than assumed, differentiates numerator and denominator separately instead of as a quotient, repeats the rule while the form still qualifies, rewrites products and differences into quotients, and reads the result as a limit of the original expression.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: applying L'Hospital's Rule to a form that is not indeterminate, or as though it were the quotient rule, or stopping while the form still qualifies, and misreading the resulting number as a function value or a derivative.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.